What exactly is “なの” (nano)?As far as I know, dropping the な is a feminine speech form. It makes it sound softer. It's the same as dropping a だ at the end of a sentence. But adding a な like in 何をしているなの sounds strange.

What does the final て in 待ってて signify?Nope, not in this situation. 待て is the imperative (like 食べろ or 飲め), it's a command to wait. It's too direct/impolite to use as a replacement for 待って unless you're commanding the other person to wait.